A railway wagon, moving with velocity v, strikes a viscously damped buffer. The motion of the wagon can be represented by the following differential equation: m = -cv v(0) = v. where m=5,000 kg is the mass of the wagon, c = 15,000 N-s/m is the damping coefficient, and the initial velocity (at 1 = 0) is vo = 3 m/s. The analytical solution of this initial value problem is: v(t) = 3e . One approach for solving this problem numerically is using the Euler's method, where the first derivative is approximated by the finite difference equation: dv Av Vi+1 - v dt At At Substituting the derivative in the ode yields the following iterative equation: Vi+1 = v; - Vat a) Write a MATLAB script for solving the differential equation using the Euler's method, from 1 = 0 to 2 seconds, using a step size of At = 0.1 s. b) Plot the results obtained in parts (a) together with the true solution. Include axis titles, plot title, gridlines and legend. Problem Two: Consider the following Gauss-Legendre algorithm for computing : a) Set a = 1, b = tant = and x = 1. b) Repeat the following commands until the difference between a and b is within some desired accuracy, E.(i.e. la- SE): y = a la- SE . . /22 Dart by t - t -x y - a) 2 2 b = by t-t-x(y-a) x 2x c) From the resulting values of a, b and I, an estimate of is: (a + b)2 plest= 41 Write a script that implements the algorithm, which is also illustrated in the flowchart. Use the while structure, and E-le-8. pi_es((a+by2)(40)